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Dimension reduction is an important tool for analyzing high-dimensional data. The predictor envelope is a method of dimension reduction for regression that assumes certain linear combinations of the predictors are immaterial to the…
In this paper, we study large and moderate deviation principles for stochastic partial differential equations (SPDEs) on metric graphs and their associated multiscale models via the weak convergence approach, providing a refined…
Semidefinite programming (SDP) problems are challenging to solve because of their high dimensionality. However, solving sparse SDP problems with small tree-width are known to be relatively easier because: (1) they can be decomposed into…
We propose the position-based scaled gradient (PSG) that scales the gradient depending on the position of a weight vector to make it more compression-friendly. First, we theoretically show that applying PSG to the standard gradient descent…
Semi-Definite Programming (SDP) with low-rank prior has been widely applied in Non-Rigid Structure from Motion (NRSfM). Based on a low-rank constraint, it avoids the inherent ambiguity of basis number selection in conventional base-shape or…
While semidefinite programming (SDP) problems are polynomially solvable in theory, it is often difficult to solve large SDP instances in practice. One technique to address this issue is to relax the global positive-semidefiniteness (PSD)…
We achieve new results on skew polynomial rings and their quotients, including the first explicit example of a skew polynomial ring where the ratio of the degree of a skew polynomial to the degree of its bound is not extremal. These methods…
Pseudospectral analysis is fundamental for quantifying the sensitivity and transient behavior of nonnormal matrices, yet its computational cost scales cubically with dimension, rendering it prohibitive for large-scale systems. While…
The coefficients in a second order parabolic linear stochastic partial differential equation (SPDE) are estimated from multiple spatially localised measurements. Assuming that the spatial resolution tends to zero and the number of…
Many high-dimensional uncertainty quantification problems are solved by polynomial dimensional decomposition (PDD), which represents Fourier-like series expansion in terms of random orthonormal polynomials with increasing dimensions. This…
This paper proposes several approaches as baselines to compute a shared active subspace for multivariate vector-valued functions. The goal is to minimize the deviation between the function evaluations on the original space and those on the…
Representing images and videos with Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, has been shown to yield high discriminative power in many visual recognition tasks.…
Mirror descent (MD) is a powerful first-order optimization technique that subsumes several optimization algorithms including gradient descent (GD). In this work, we develop a semi-definite programming (SDP) framework to analyze the…
Semidefinite programming (SDP) is a fundamental class of convex optimization problems with diverse applications in mathematics, engineering, machine learning, and related disciplines. This paper investigates the application of the…
Positive semidefinite (PSD) matrices are indispensable in many fields of science. A similarity measurement for such matrices is usually an essential ingredient in the mathematical modelling of a scientific problem. This paper proposes a…
In this paper, we propose a dimension reduction model for spatially dependent variables. Namely, we investigate an extension of the \emph{inverse regression} method under strong mixing condition. This method is based on estimation of the…
The coefficient function of the leading differential operator is estimated from observations of a linear stochastic partial differential equation (SPDE). The estimation is based on continuous time observations which are localised in space.…
Consider a convex set S defined by a matrix inequality of polynomials or rational functions over a domain. The set S is called semidefinite programming (SDP) representable or just semidefinite representable if it equals the projection of a…
The purpose of this paper is to show and explain a new formula that indicates with finality the derivatives of Shifted Monic Ultraspherical polynomials (SMUPs) of any degree and for any fractional-order using the shifted Monic…
This paper considers the problem of positive semidefinite factorization (PSD factorization), a generalization of exact nonnegative matrix factorization. Given an $m$-by-$n$ nonnegative matrix $X$ and an integer $k$, the PSD factorization…